Most of the serious large scale carbon capture projects is doing exactly this. But this comes with its own issues. When you burn coal or oil you do not just get carbon dioxide but also lots of sot, tar, acids, radioactive isotopes, and tons of water vapor. This damages the filters used for capturing the carbon dioxide.
There are other chemical processes which produce more pure carbon dioxide. Most notably cement production does output almost pure carbon dioxide from one of their processes. This is a major contributor to carbon dioxide emission with over 30% of the worlds emissions coming from cement production. It is hard to retrofit an existing cement plant with carbon capture technology as they do not collect the gas from the process and also tends to mix it with gas from burning coal or oil which is used as a heat source for the process. But new cement plant is built to capture the pure carbon dioxide and make it easily available for carbon capture.
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